4 série, t. 35, 2002, p. 1 à 26.
FINITE QUANTUM GROUPS OVER ABELIAN GROUPS OF PRIME EXPONENT
B
YN
ICOLÁSANDRUSKIEWITSCH
ANDH
ANS-J
ÜRGENSCHNEIDER
ABSTRACT. – We classify pointed finite-dimensional complex Hopf algebras whose group of group-like elements is abelian of prime exponentp, p >17. The Hopf algebras we find are members of a general family of pointed Hopf algebras we construct from Dynkin diagrams. As special cases of our construction we obtain all the Frobenius–Lusztig kernels of semisimple Lie algebras and their parabolic subalgebras. An important step in the classification result is to show that all these Hopf algebras are generated by group-like and skew-primitive elements.
2002 Éditions scientifiques et médicales Elsevier SAS
RÉSUMÉ. – Nous classifions les algèbres de Hopf complexes de dimension finie dont le groupe des éléments groupoïdaux est abélien d’exposant premierp,p >17. Les algèbres de Hopf que nous trouvons appartiennent à une famille d’algèbres de Hopf pointées que nous construisons à partir de diagrammes de Dynkin. Comme cas particuliers de notre construction nous obtenons tous les noyaux de Frobenius–
Lusztig des algèbres de Lie semi-simples et leurs sous-algèbres paraboliques. Une étape importante dans notre classification consiste à montrer que toutes ces algèbres de Hopf sont engendrées par des éléments groupoïdaux et des éléments primitifs tordus.
2002 Éditions scientifiques et médicales Elsevier SAS
1. Introduction
Since the discovery of quantum groups (Drinfeld, Jimbo) and their finite dimensional variations (Lusztig, Manin), these objects were studied from different points of view and had many applications. The present paper is part of a series where we intend to show that important classes of Hopf algebras are quantum groups and therefore belong to Lie theory.
We will assume that the ground-fieldkis algebraically closed of characteristic0. One of our main results is the explicit construction of a general family of pointed Hopf algebras from Dynkin diagrams (Theorem 5.17). All the Frobenius–Lusztig kernels and their parabolic subalgebras belong to this family, but in addition we get many new examples. We show that any finite dimensional pointed Hopf algebra with group of prime exponent (greater than 17) is indeed in this family; see our main Theorem below. An important step in the proof follows from another main result (Theorem 7.6), where we show that a large family of finite dimensional pointed Hopf algebras is generated by group-like and skew-primitive elements, giving additional support to a conjecture in [5].
IfAis a Hopf algebra, then we denote byG(A)the group of group-like elements ofA. Recall thatAis pointed ifkG(A)is the largest cosemisimple subcoalgebra ofA, or equivalently if any irreducibleA-comodule is one-dimensional.
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE
LetΓbe a finite abelian group andΓthe group of its characters. We denote the unit element inΓbyε.
To state our main result, we have to introduce some notation. A linking datumD of finite Cartan type forΓis a collectionDconsisting of
a Cartan matrix of finite type (aij)1i,jθ of size θ×θ for some θ1, [20], elements g1, . . . , gθ∈Γ, charactersχ1, . . . , χθ∈Γsatisfying
χi(gi)= 1, for all1iθ, (1.1)
χj(gi)χi(gj) =χi(gi)aij, for all1i, jθ, (1.2)
and a family (λij)1i<jθ, ij of elements in k such that λij is arbitrary if gigj = 1 and χiχj=ε, but 0 otherwise.
The elements(λij)1i<jθ, ijare called the linking elements ofD.
Here, byi∼j, resp.ij,1i, jθwe mean thatiandj belong to the same connected component, resp. to different connected components of the Dynkin diagram corresponding to (aij).
Now we fix a primep >2and a natural numbers. We consider finite abelian groups of the formΓ(s) := (Z/(p))s.
LetDbe a linking datum of finite Cartan type forΓ(s)with Cartan matrix(aij)1i,jθ and linking elements(λij)1i<jθ, ij.
We define the algebrau(D)by generatorsa1, . . . , aθ,y1, . . . , ysand relations yph= 1, ymyh=yhym, for all1m, hs, (1.3)
gaj=χj(g)ajyh, for allg∈Γ, 1jθ, (1.4)
(adai)1−aijaj= 0, for all1i=jθ, i∼j, (1.5)
aiaj−χj(gi)ajai=λij(1−gigj), for all1i < jθ, ij;
(1.6)
apα= 0, for allα∈Φ+. (1.7)
To formulate these relations we used the following natural interpretation of elements g∈Γ as words in the generatorsyh,1hs. LetYh,1hs, be aZ/(p)-basis ofΓ, and write g=Y1t1· · ·Ysts, wheret1, . . . , tsare natural numbers. Then in the relations above replacegby the formal expressionyt11· · ·ytss.
In (1.5),adaiis the adjoint action ofai, that is for allx∈u(D), (adai)x=ai(1)xS(ai(2)) =aix−gixgi−1ai.
In this way the left hand side of (1.5) is meant as a well-defined expression in the generators.
In (1.7),Φ+is the set of positive roots of the root system associated to the Cartan matrix(aij);
the “root vectors”aαare defined in Section 4.1 below.
Our main theorem is
THEOREM 1.8. – (a) LetDbe a linking datum of finite Cartan type forΓ(s)with Cartan matrix(aij), and assume thatp >3if(aij)has a connected component of typeG2. Thenu(D) has a unique Hopf algebra structure determined by
∆yh=yh⊗yh, ∆ai=ai⊗1 +gi⊗ai, for all1hs, 1iθ.
(1.9)
The Hopf algebrau(D)is pointed,G(u(D))Γ(s)anddimu(D) =ps+|Φ+|.
(b) Letp >17. LetAbe a pointed finite-dimensional Hopf algebra such thatG(A)Γ(s).
Then there exists a linking datumDof finite Cartan type forΓ(s)such thatA u(D).
Remarks 1.10. – (i) In Section 5 we define the notion of a “linking datum” for a general finite abelian groupΓ. In the situation of the main theorem it is always possible to reduce to the case of linking data with all entriesλijequal to0or1. Thus it follows from Theorem 1.8 that there are only finitely many isomorphism classes of finite dimensional Hopf algebras with fixed coradical kΓ(s). For more general finite abelian groups, this is no longer true [4,8,16].
(ii) The dimensions of the Hopf algebras in Theorem 1.8 are very special numbers. This phenomenon is shown in general for arbitrary finite groupsΓin Theorem 7.9.
(iii) Let (aij)1i,jθ be a finite Cartan matrix. The problem of actually finding all the collectionsgi∈Γ(s),χi∈Γ(s),1iθ, such that (1.1) and (1.2) hold has been discussed in [5]. It can be stated as the problem of finding all the solutions of a system of algebraic equations overZ/(p)and it is in principle solvable. Note that in particular
θ2sp−1 p−2, see [5, Prop. 8.3].
(iv) The question of finding all the possible linking elements attached to a fixed collection gi,χi,1iθ,(aij)1i,jθ, is also of combinatorial nature, see Section 5, and also [13]. Once these two problems are solved effectively, the isomorphism classes of the Hopf algebrasu(D) can be determined using [5, Prop. 6.3], [6, Lemma 1.2].
(v) As a consequence of Theorem 1.8 one obtains the complete classification of all finite dimensional pointed Hopf algebras with group of group-likesΓ(1) =Z/(p),p= 5,7. It is the list given in [5, Theorem 1.3] plus the Frobenius–Lusztig kernels as described in [4]. Indeed, replacing in the proof of Theorem 1.8 [5, Cor. 1.2] by [5, Th. 1.3] we get the classification for all primesp= 5or7, in view of Theorem 6.8 and [6, Lemma 4.2]. The only cases not covered arep= 5, typeB2andp= 7, typeG2. This result was independently obtained by Musson [28]
using different methods starting from our previous article [5].
(vi) Up to now, the determination of all finite dimensional pointed Hopf algebras A with G(A)Γ, for a fixed groupΓ, was known only forΓ =Z/(2)[29]. Other classification results of pointed Hopf algebras are known for some fixed dimensiond:d=p2is easy and follows from [29,30];d=p3was done in [4], and by different methods in [10,36];d=p4in [6] (and does not seem to be possible via the other methods);d= 16in [11],d= 32in [17]; results on the case whenΓhas exponent2can be found in [2].
(vii) The classification of all coradically graded pointed Hopf algebras of dimensionp5was obtained in [18]. It is not difficult to deduce the classification of all pointed Hopf algebras of dimensionp5using Theorem 1.8 and results in [6].
(viii) The Hopf algebras u(D) can be defined for any Cartan datum of finite type D of an arbitrary finite abelian group. Part (a) of Theorem 1.8 is a special case of the general Theorem 5.17. For suitable choices of D, the Frobenius–Lusztig kernels and their parabolic subalgebras are of the formu(D). See Example 5.12. Otherwise Theorem 5.17 provides many new examples of finite dimensional Hopf algebras arising from exotic linking data.
(ix) The results of this paper heavily depend on our paper [5] and on previous work on quantum groups [21,22,33,34,12,27].
Conventions. Our reference for the theory of Hopf algebras is [26]. The notation for Hopf algebras is standard:∆,S,, denote respectively the comultiplication, the antipode, the counit;
we use Sweedler’s notation but dropping the summation symbol.
IfCis a coalgebra thenG(C)denotes the set of its group-like elements andC0⊂C1⊂ · · · its coradical filtration. So that C0 is the coradical ofC. As usual,Pg,h(C)denotes the space
of(g, h)-skew primitives ofC,g, h∈G(C). IfChas a distinguished group-like 1, then we set P(C) =P1,1(C), the space of primitive elements ofC.
IfA is an algebra and(xi)i∈I is a family of elements ofAthenkxii∈I or simplykxi, resp.xii∈I orxidenotes the subalgebra, resp. the two-sided ideal, generated by thexi’s.
LetHbe a Hopf algebra. A Yetter–Drinfeld module overHis a vector spaceV provided with structures of leftH-module and leftH-comodule such thatδ(h.v) =h(1)v(−1)Sh(3)⊗h(2).v(0). We denote byHHYDthe (braided) category of Yetter–Drinfeld modules overH.
Assume thatH=kΓwhereΓis a finite abelian group. We denoteΓΓYD:=HHYD. Letg∈Γ, χ∈ΓandV a module, resp. a comodule, resp. a Yetter–Drinfeld module, overΓ. Then we denote Vχ={v∈V: h.v=χ(h)v,∀h∈Γ}, resp.Vg={v∈V: δ(v) =g⊗v}, resp.Vgχ:=Vg∩Vχ. IfV is a locally finite Yetter–Drinfeld module, then V =
g∈Γ,χ∈ΓVgχ. Conversely, a vector spaceV provided with a direct sum decompositionV =
g∈Γ,χ∈ΓVgχ has an evident Yetter–
Drinfeld module structure.
2. Outline of the paper and proof of the main result
Theorem 1.8 follows from Theorems 4.5, 5.17, 6.8, 6.10 and Corollary 7.7 in the present article, along the guidelines proposed in [4]. We give now the proof of Theorem 1.8 assuming those results which hold over arbitrary finite abelian groups. This section serves also as a guide to the different sections of the paper.
2.1. The proof
LetAbe a finite dimensional pointed Hopf algebra withG(A)Γ(s). Let grA:=
n0
grA(n),
where grA(0) =A0, grA(n) =An/An−1, if n >0 be the graded coalgebra associated to the coradical filtration of A. ThengrA is a graded Hopf algebra [26] and both the inclusion ι:A0→grAand the graded projectionπ: grA → A0are Hopf algebra maps. Let
R:= grAcoπ=
x∈grA: (id⊗π)∆(x) =x⊗1
;
it is a graded braided Hopf algebra inΓΓ(s)(s)YDwith the grading inherited fromgrA:
R=
n0
R(n), R(n) :=R∩grA(n).
Notice thatgrAcan be reconstructed fromRas a biproduct:
grA R#kΓ(s).
The braided Hopf algebraRis called the diagram ofA. One has R(0) =k1,
(2.1)
R(1) =P(R), (2.2)
and we know from Corollary 7.7 below that
Ris generated as an algebra by R(1).
(2.3)
LetV :=R(1); it is a Yetter–Drinfeld submodule ofR. SinceRsatisfies (2.1), (2.2) and (2.3) we know thatRB(V)is a Nichols algebra, see Section 3.2. Now there exists a basisx1, . . . , xθ
ofV andg1, . . . , gθ∈Γ(s),χ1, . . . , χθ∈Γ(s)such thatxi∈Vgχii,1iθ. SinceAis finite dimensional,χi(gi)= 1for alli[4, Lemma 3.1] and there is a finite Cartan matrix(aij)1i,jθ
such that (1.2) holds [5, Cor. 1.2].
To give an explicit description ofB(V), we introduce root vectors inB(V)generalizing the root vectors defined in [21]. We note that Lusztig’s root vectors can be described up to a non- zero scalar as an iterated braided commutator of simple root vectors. We then define the root vectors in the general case by exactly the same iterated braided commutator with respect to our more general braiding. As one of our main results, we obtain a presentation by generators and relations and a PBW basis forB(V)from the corresponding Theorem for Frobenius–Lusztig kernels, using Drinfeld’s twisting essentially in the same way as in [5]. See Theorem 4.5. We can then deduce part (a) of Theorem 1.8. For connected Dynkin diagrams it is a consequence of Theorem 4.5; in the non-connected case we apply the idea of twisting the algebra structure by a 2-cocycle which is given by a Hopf algebra map [14]. See Theorem 5.17.
It follows from Theorem 4.5 thatgrAcan be presented as an algebra by generatorsy1, . . . , ys
(homogeneous of degree0) andx1, . . . , xθ(homogeneous of degree1), and relations yph= 1, ymyh=yhym, for all1m, hs, (2.4)
yhxj=χj(yh)xjyh, for all1hs, 1jθ, (2.5)
(adxi)1−aijxj= 0, for all1i=jθ;
(2.6)
xpα= 0, for allα∈Φ+; (2.7)
and where the Hopf algebra structure is determined by
∆yh=yh⊗yh, ∆xi=xi⊗1 +gi⊗xi, for all1hs, 1iθ.
(2.8)
By [4, Lemma 5.4], we can choose ai∈ Pgi,1(A)χi such that the class of ai in grA(1) coincides with xi. We also keep the notation yj for the generators of G(A). It is clear that relations (1.3) and (1.4) hold. Now relations (1.5) and (1.6), resp. (1.7), hold because of Theorem 6.8, resp. Lemma 6.9.
The Theorem now follows from Theorem 6.10. 2 2.2. The general case
There are several obstructions to extend Theorem 1.8 to general finite abelian groups. First, it is open whether the diagram of a finite dimensional pointed Hopf algebra is generated in degree one, i.e. when it is a Nichols algebra; second, there are finite dimensional Nichols algebras which are not of Cartan type [29].
For liftings ofgrAwhenRis a Nichols algebra of Cartan type, the quantum Serre relations of connected vertices in general still hold as we show in Theorem 6.8 below; however the powers of the root vectors are not necessarily0. We should haveaNαα=uα∈kG(A); the determination ofuαwhenαis a non-simple root was done in [6] for typeA2, in [9] for typeB2, and in [7] for typeAnfor anyn(up to some exceptional cases concerning the orders of the roots of unity).
3. Braided Hopf algebras 3.1. Biproducts
LetR be a braided Hopf algebra inHHYD; this means thatRis an algebra and a coalgebra inHHYDand that the comultiplication∆R:R→R⊗R is an algebra map when inR⊗Rthe multiplication twisted by the braiding c is considered; in additionR admits an antipode. To avoid confusions we use the following variant of Sweedler’s notation for the comultiplication of R:∆R(r) =r(1)⊗r(2). LetA=R#H be the biproduct or bosonization ofR[24], [31]. Recall that the multiplication and comultiplication ofAare given by
(r#h)(s#f) =r(h(1).s)#h(2)f, ∆(r#h) =r(1)#(r(2))(−1)h(1)⊗(r(2))(0)#h(2). The maps π:A →H and ι:H →A, π(r#h) =(r)h, ι(h) = 1#h, are Hopf algebra homomorphisms; we have R={a∈A: (id⊗π)∆(a) =a⊗1}. Conversely, let A, H be Hopf algebras provided with Hopf algebra homomorphismsπ:A→H and ι:H →A. Then R={a∈A: (id⊗π)∆(a) =a⊗1}is a braided Hopf algebra inHHYD. The action.ofH on Ris the restriction of the adjoint action (composed withι) and the coaction is(π⊗id)∆;Ris a subalgebra ofAand the comultiplication is∆R(r) =r(1)ιπS(r(2))⊗r(3). These constructions are inverse to each other. We shall mostly omitιin what follows.
Letϑ:A→Rbe the map given byϑ(a) =a(1)πS(a(2)). Then ϑ(ab) =a(1)ϑ(b)πS(a(2)), (3.1)
for all a, b∈A and ϑ(h) =ε(h) for all h∈H; therefore, for all a∈A, h∈H, we have ϑ(ah) =ϑ(a)ε(h)and
ϑ(ha) =h.ϑ(a) =ϑ
h(1)aπS(h(2)) . (3.2)
Notice also that ϑinduces a coalgebra isomorphismA/AH+R. In fact, the isomorphism A→R#Hcan be expressed explicitly as
a→ϑ(a(1))#π(a(2)), a∈A.
IfA is a Hopf algebra, the well-known adjoint representationadofAon itself is given by adx(y) =x(1)yS(x(2)). IfR is a braided Hopf algebra in HHYDthen there is also a braided adjoint representationadcofRon itself given by
adcx(y) =µ(µ⊗ S)(id⊗c)(∆⊗id)(x⊗y),
whereµis the multiplication andc∈End(R⊗R)is the braiding. Note that ifx∈ P(R)then the braided adjoint representation ofxis just
adcx(y) =µ(id−c)(x⊗y) =: [x, y]c. (3.3)
The element[x, y]c defined by the second equality for anyxandy, regardless of whetherxis primitive, will be called a braided commutator.
WhenA=R#H, then for allb, d∈R,
ad(b#1)(d#1) =
adcb(d)
#1.
(3.4)
3.2. Nichols algebras
LetHbe a Hopf algebra and letR=
n∈NR(n)be a graded braided Hopf algebra inHHYD. We say thatRis a Nichols algebra if 2.1, 2.2 and 2.3 hold, cf. [29,5,3]. A Nichols algebraRis uniquely determined by the Yetter–Drinfeld moduleP(R); given a Yetter–Drinfeld moduleV, there exists a unique (up to isomorphism) Nichols algebraRwithP(R)V. It will be denoted B(V). In fact, the kernel of the canonical map:T(V)→B(V)can be described in several different ways. For instance,Ker=
n0KerSn whereSn is the “quantum symmetrizer”
defined from the braidingc; so that B(V) is a “quantum shuffle algebra” and as algebra and coalgebra only depends on the braidingc:V⊗V →V ⊗V. See [29,37,23,33–35].
Let H =kΓ where Γ is a finite abelian group. Let V be a finite dimensional Yetter–
Drinfeld module overΓ. Then there exist a basisx1, . . . , xθofV and elementsg1, . . . , gθ∈Γ, χ1, . . . , χθ∈Γsuch that
xj∈Vgχjj, for all1jθ.
(3.5)
In what follows we shall only consider Yetter–Drinfeld modules V such that χi(gi)= 1, 1iθ. The braidingcis given with respect to the basisxi⊗xjbyc(xi⊗xj) =bijxj⊗xi, where
(bij)1i,jθ= χj(gi)
1i,jθ.
Remark 3.6. – LetV, resp.V, be a finite dimensional Yetter–Drinfeld module overΓ, resp.
Γ, with a basis x1, . . . , xθsuch thatxi∈Vgχii, resp. with a basisx˜1, . . . ,x˜θ such thatx˜i∈V˜gχ˜ii. Assume that χi(gj) = ˜χi(˜gj) for all 1i, j θ. Then there exists a unique algebra and coalgebra isomorphismB(V)→B(V)such thatxi→x˜ifor all1iθ.
DEFINITION 3.7. – We shall say that a braiding given by a matrixb= (bij)1i,jθ whose entries are roots of unity is of Cartan type if for alli, j,bii= 1and there existsaij∈Zsuch that
bijbji=baiiij.
The integersaij are uniquely determined by the following rules:
• Ifi=jwe takeaii= 2;
• ifi=j, we select the uniqueaijsuch that−ordbii< aij0.
Then(aij)is a generalized Cartan matrix [20]. We shall say a Yetter–Drinfeld moduleV is of Cartan type, resp. finite Cartan type, if its corresponding braiding is of Cartan type, resp. the same plus the matrix(aij)is of finite type.
3.3. The twisting functor
LetH be a Hopf algebra and letF be an invertible element inH⊗H such that F12(∆⊗id)F=F23(id⊗∆)F, (ε⊗id)(F) = 1 = (id⊗ε)(F).
(3.8)
ThenHF, the same algebraH but with the comultiplication∆F:=F∆F−1, is again a Hopf algebra [15]. We shall writeF=F1⊗F2,F−1=G1⊗G2; the new comultiplication will be denoted by∆F(h) =h(1,F)⊗h(2,F).
Let nowRbe a braided Hopf algebra inHHYD, letA=R#Hbe its bosonization and consider the Hopf algebra AF. It follows from the definitions thatπ:AF →HF andι:HF →AF are
also Hopf algebra homomorphisms. Hence
RF:={a∈AF: (id⊗π)∆F(a) =a⊗1}
is a braided Hopf algebra in the categoryHHF
FYD. We consider the corresponding mapϑF and defineψ:R→RF by
ψ(r) =ϑF(r), r∈R.
(3.9)
The mapψwas defined in [5] in the caseH=kΓis the group algebra of a finite abelian group.
The following lemma generalizes [5, Lemma 2.3]; part (iii), new even forH=kΓ, will be needed in the sequel.
LEMMA 3.10. – (i)ψis an isomorphism ofH-modules. (Recall thatH=HF as algebras.) (ii) Ifr, s∈Rthen
ψ(rs) =F1.ψ(r)F2.ψ(s).
(3.11)
(iii) Ifr∈Rthen
∆RFψ(r) =F1.ψ r(1)
⊗F2.ψ r(2)
. (3.12)
(iv) IfRis a graded braided Hopf algebra, thenRF also is andψis a graded map. IfRis a coradically graded braided Hopf algebra (resp. a Nichols algebra), thenRF also is.
Proof. – (i) follows from (3.2):ψ(h.r) =ϑF(h.r) =ϑF(hr) =h.ϑF(r) =h.ψ(r).Now we prove (ii):
ψ(rs) =ϑF(rs) =r(1,F)ϑF(s)π
SF(r(2,F))
=r(1,F)π
SF(r(2,F))
π(r(3,F))ϑF(s)π
SF(r(4,F))
=ψ(r(1,F))π(r(2,F)).ψ(s) =ψ
F1r(1)G1 π
F2r(2)G2 .ψ(s)
=ψ F1r(1)
ε G1
π
F2r(2)G2
.ψ(s) =F1.ψ(r(1))π
F2)π(r(2)
.ψ(s)
=F1.ψ(r)π(F2).ψ(s),
as claimed. Here we have used (3.1), the definitions and (3.8). For the proof of (iii), we first observe that, ifr∈R, then
ψ r(1)
⊗ψ r(2)
=ϑF
r(1)πS(r(2))
⊗ϑF(r(3)) =ψ(r(1))⊗ψ(r(2)).
(3.13)
Using thatϑF is a coalgebra map, (3.2) and (3.13), we conclude that
∆RFψ(r) = ∆RFϑF(r) =ϑF(r(1,F))⊗ϑF(r(2,F))
=ϑF
F1r(1)G1
⊗ϑF
F2r(2)G2
=ϑF
F1r(1)
⊗ϑF
F2r(2)
=F1.ϑF(r(1))⊗F2.ϑF(r(2)) =F1.ψ r(1)
⊗F2.ψ r(2)
.
The proof of (iv) has no difference with the proof of the analogous statement in [5, Lemma 2.3]. 2
We now consider the special case whenH=kΓ,Γa finite abelian group. Letω:Γ×Γ→k× be a2-cocycle, i.e.ω(τ,1) =ω(1, τ) = 1andω(τ, ζ)ω(τ ζ, η) =ω(τ, ζη)ω(ζ, η). The cocycleω allows to define a mapΨ :Γ×Γ→Γby
τ,Ψ(χ, g)=ω(τ, χ)ω(χ, τ)−1τ, g, τ∈Γ.
(3.14)
We identifyH with the Hopf algebrakΓ of functions on the groupΓ; we denote by δτ ∈H the function given byδτ(ζ) =δτ,ζ,τ, ζ∈Γ. Thenδτ=|Γ1| g∈Γτ, g−1g.LetF∈H⊗H be given by
F=
τ,ζ∈Γ
ω(τ, ζ)δτ⊗δζ.
ThenFsatisfies (3.8); note thatH=HF. Let nowRbe a braided Hopf algebra inΓΓYD; we can consider the Hopf algebrasA=R#kΓandAF, the braided Hopf algebraRF∈ΓΓYDand the mapψ:R→RF. We have
ψ(r) =
τ∈Γ
ω(χ, τ)−1r#δτ, r∈Rχ; ψ Rχg
=RψΨ(χ,g).
See [5, Lemma 2.3]. Note that (3.11) is nowψ(rs) =ω(χ, τ)ψ(r)ψ(s),r∈Rχ,s∈Rτ. LEMMA 3.15. – Ifr∈P(R)χg ands∈Rτthen
ψ [r, s]c
=ω(χ, τ)[ψ(r), ψ(s)]c. (3.16)
Proof. – We have ψ
[r, s]c
=ψ
rs−τ(g)sr
=ω(χ, τ)ψ(r)ψ(s)−ω(τ, χ)τ(g)ψ(s)ψ(r)
=ω(χ, τ)
ψ(r)ψ(s)− τ,Ψ(χ, g)ψ(s)ψ(r)
=ω(χ, τ)[ψ(r)), ψ(s)]c, where we used (3.14). 2
Remark 3.17. – It is possible to show that (ψ⊗ψ)c(r⊗s) =F.cF(ψ(r)⊗ψ(s)), for all r∈Rχg,s∈Rτ.
From the previous considerations and Lemma 3.10 we immediately get
PROPOSITION 3.18. – Let R be an algebra in ΓΓYD, (xi)i∈I a family of elements of R, xi∈Rgχiifor somegi∈Γ,χi∈Γ. Then:
(i) ψ(kxi) =kψ(xi),ψ(xi) =ψ(xi).
(ii) If R has a presentation by generators xi and relations tj, where also the tj’s are homogeneous thenRF has a presentation by generatorsψ(xi)and relationsψ(tj).
(iii) Ifxiis central andω(χi, τ) =ω(τ, χi)for allτsuch thatRτ= 0, thenψ(xi)is central.
4. Root vectors and Quantum Serre relations 4.1. Root vectors
In this section, we assume the following situation:
We fix a finite abelian group Γ, a finite Cartan matrix (aij)1i,jθ and g1, . . . , gθ∈Γ, χ1, . . . , χθ∈Γsuch that (1.1) and (1.2) hold. Letd1, . . . , dθ∈ {1,2,3}such thatdiaij=djaji
for alli,j. We setqi=χi(gi),Nithe order ofqi. We assume, for alliandj, that the order of χi(gj)is odd, and thatNiis not divisible by3ifibelongs to a connected component of typeG2.
Let X be the set of connected components of the Dynkin diagram corresponding to (aij).
We assume that for eachI∈ X, there existcI, dI such thatI={j: cI jdI}; that is, after reordering the Cartan matrix is a matrix of blocks corresponding to the connected components.
Let I∈ X and i∼j in I; then Ni =Nj, hence NI :=Ni is well defined. Let ΦI, resp.
Φ+I, be the root system, resp. the subset of positive roots, corresponding to the Cartan matrix (aij)i,j∈I; thenΦ =
I∈XΦI, resp. Φ+ =
I∈XΦ+I is the root system, resp. the subset of positive roots, corresponding to the Cartan matrix (aij)1i,jθ. Let α1, . . . , αθ be the set of simple roots.
LetWI be the Weyl group corresponding to the Cartan matrix(aij)i,j∈I; we identify it with a subgroup of the Weyl group W corresponding to the Cartan matrix(aij). We fix a reduced decomposition of the longest elementω0,IofWI in terms of simple reflections. Then we obtain a reduced decomposition of the longest elementω0=si1. . . siP ofWfrom the expression ofω0
as product of theω0,I’s in some fixed order of the components, say the order arising from the order of the vertices. Thereforeβj:=si1. . . sij−1(αij)is a numeration ofΦ+.
We fix a finite dimensional Yetter–Drinfeld moduleV overΓ with a basisx1, . . . , xθ with xi∈Vgχii,1iθ.
Major examples of modules of Cartan type are the Frobenius–Lusztig kernels. LetN >1be an odd natural number and letq∈kbe a primitiveNth root of1, not divisible by3in case(aij) has a component of type G2. LetG=Z/(N)θ=e1 ⊕ · · · ⊕ eθ; letηj∈G be the unique character such thatη(j), e(i)=qdiaij. LetVbe a Yetter–Drinfeld module overGwith a basis X1, . . . , Xθsuch that
Xi∈Veηii, for all1iθ.
We denote by c the braiding of V. Lusztig defined root vectors Xα∈B(V), α∈Φ+ [22].
One can see from [23] that, up to a non-zero scalar, each root vector can be written as an iterated braided commutator in some sequence X1, . . . , Xa of simple root vectors such as [[X1,[X2, X3]c]c,[X4, X5]c]c. This can also be seen in the situation in [32].
We now fix for eachα∈Φ+such a representation ofXαas an iterated braided commutator. In the general case of ourV, we define root vectorsxαin the tensor algebraT(V),α∈Φ+, as the same formal iteration of braided commutators in the elementsx1, . . . , xθinstead ofX1, . . . , Xθ
but with respect to the braidingc given by the general matrix (χj(gi)). Note that each xα is homogeneous and has the same degree asXα, where we mean the degree in the sense of [23].
Also,
xα∈T(V)χgαα, (4.1)
wheregα=g1b1· · ·gbθθ,χα=χb11· · ·χbθθ, whereα=b1α1+· · ·+bθαθ.
THEOREM 4.2. – The Nichols algebraB(V)is presented by generatorsXi,1iθ, and relations
adc(Xi)1−aij(Xj) = 0, for alli=j, (4.3)
XαN= 0, for allα∈Φ+. (4.4)
Moreover, the following elements constitute a basis ofB(V):
Xβh11Xβh22· · ·XβhPP, for all0hjN−1, 1jP.
Proof. – It follows from results of Lusztig [21,22], Rosso [33,34] and Müller [27] thatB(V) is the positive part of the so-called Frobenius–Lusztig kernel corresponding to the Cartan matrix (aij). See [5, Th. 3.1] for details. The presentation by generators and relations follows from the
considerations in the last paragraph of p. 15 and the first paragraph of p. 16 in [1] referring to [12, §19, Corollary in p. 120]. The statement about the basis is [21,22]. 2
4.2. Nichols algebras of Cartan type
We can now prove the first main result of the present paper, describingB(V)by generators and relations whenV is of finite Cartan type, improving [5, Th. 1.1(i)]. As in loc. cit., we use repeatedly Remark 3.6.
THEOREM 4.5. – The Nichols algebraB(V)is presented by generatorsxi,1iθ, and relations
adc(xi)1−aij(xj) = 0, for alli=j, (4.6)
xNαI= 0, for allα∈Φ+I, I∈ X. (4.7)
Moreover, the following elements constitute a basis ofB(V):
xhβ1
1xhβ2
2· · ·xhβP
P, for all0hjNI−1,ifβj∈I, 1jP.
Proof. – (a) Let us first assume that the braiding is symmetric, that isχi(gj) =χj(gi)for all i, j. By [5, Lemma 4.2] we can assume moreover that the Cartan matrix(aij)is connected. From our assumptions on the orders of theχi(gj)we then conclude that the braiding has the form χj(gi) =qdiaij for alli, jwhereqis a root of unity of orderN=χi(gi). See [5, Lemma 4.3].
Hence the Theorem follows directly from Theorem 4.2 and Remark 3.6.
(b) In the case of an arbitrary braiding we know from Lemma 4.1 of [5] that there exists a finite abelian groupGsatisfying:
• The braidingcofV can be realized from a Yetter–Drinfeld module structure overGthat we continue denoting byV, cf. Remark 3.6.
• There exists a cocycleω:G ×G →k× with correspondingF ∈kG⊗kG such that the braiding ofVF is symmetric. Letψ:B(V)→B(VF)be the isomorphism having the same meaning as in (3.9).
• The braiding of VF is given in the basis ψ(xi)⊗ψ(xj) by a matrix (bFij) such that bFii=χi(gi)and the order of(bFij)is again odd for alliandj.
If :T(V)→B(V), F:T(VF)→B(VF) denote the canonical maps, then we have a commutative diagram
T(V)
ψ
B(V)
ψ
T(VF) F B(VF).
Clearly, ψ(Ker) = KerF; if (rj)j∈J is a set of generators of the ideal Ker with rj∈T(V)ηhj
j then by Proposition 3.18(ψ(rj))j∈J is a set of generators of the idealKerF. By the symmetric case (a), we know the generators of KerϕF. Let us denote Xi:=ψ(xi).
Then by Lemma 3.15 and (3.11), we have ψ(adc(xi)1−aij(xj)) =uijadc(Xi)1−aij(Xj) and ψ
xNαI
=uαXαNI,α∈Φ+I whereuij, uαare non-zero scalars. This implies the first claim of the theorem. The second follows in a similar way. 2
LetB(V )be the braided Hopf algebra inΓΓYDgenerated byx1, . . . , xθwith relations (4.6), where thexi’s are primitive. LetK(V)be the subalgebra ofB(V )generated byxNαI,α∈Φ+I, I∈ X; it is a Yetter–Drinfeld submodule ofB(V ).
THEOREM 4.8. – K(V)is a braided Hopf subalgebra inΓΓYDofB(V ).
Proof. – (a) As in the proof of Theorem 4.5 we first assume that the braiding is symmetric.
If i=j, then χj(gi)χi(gj) = 1 and hence the corresponding Serre relation (4.6) says that xixj=xjxi. Thus, we can easily reduce to the connected case. In such case, χj(gi) =qdiaij as before and the Theorem is shown in [12].
(b) In the general case, we change the group as in the proof of Theorem 4.5. The isomorphism ψ:T(V)→T(VF)respects the Serre relations up to non-zero scalars by Lemma 3.15. Also, it maps subcoalgebras stable under the action of the group to subcoalgebras by Lemma 3.10(iii).
We conclude from (a) thatK(V)is a subcoalgebra ofB(V ). 2
5. Linking datum and glueing of connected components 5.1. Linking datum
In this section, we fix a finite abelian group Γ, a finite Cartan matrix (aij)1i,jθ and g1, . . . , gθ∈Γ,χ1, . . . , χθ∈Γsuch that (1.1) and (1.2) hold. We preserve the conventions and hypotheses from Section 4.
DEFINITION 5.1. – We say that two verticesiandjare linkable (or thatiis linkable toj) if ij,
(5.2)
gigj= 1 and (5.3)
χiχj= 1.
(5.4)
Ifiis linkable toj, thenχi(gj)χj(gi) = 1by (5.2); it follows then from (5.4) that χj(gj) =χi(gi)−1.
(5.5)
LEMMA 5.6. – Assume thatiandk, resp.jand, are linkable. Thenaij=ak,aji=ak. In particular, a vertexican not be linkable to two different verticesjandh.
Proof. – Ifai= 0thenaij=aji= 0(otherwisej∼) andak=ak= 0(otherwisei∼k).
Ifajk= 0thenaij=aji= 0(otherwisei∼k) and ak=ak= 0(otherwisej∼). Assume thatai= 0 =ajk. Then
χi(gi)aij =χi(gj)χj(gi) =χ−k1(gj)χ−1(gi) =χj(gk)χi(g)
=χ−1(gk)χ−k1(g) =χk(gk)−ak=χi(gi)ak.
Then Ni dividesaij −ak and analogously,Nk divides aij−ak. So that aij =ak by the assumptions on the order of Ni andNk; by symmetry,aji=ak. Assume that a vertexi is linkable tojandh. Then2 =aii=ajh, soj=h. 2
A linking datum of finite Cartan type forΓis a collection D=D
Γ, (aij)1i,jθ, (gi)1iθ, (χj)1jθ,(λij)1i<jθ, ij
where (aij) is a Cartan matrix of finite type, (gi),(χj) are elements as above, and (λij)1i<jθ, ijare elements inksuch thatλij is arbitrary ifiandj are linkable but 0 other- wise. The elementsλij are called the linking elements ofD. Given a linking datum, we say that two verticesiandjare linked ifλij= 0.
This definition generalizes part of the definition of compatible datum in [4, Section 5]. We shall represent a linking datum by the Dynkin diagram of the Cartan matrix (aij) joining linked vertices by a dotted line. To have a complete picture we add the pair(gi, χi)below the vertexi.
DEFINITION 5.7. – Let us fix a decompositionΓ =Y1⊕· · ·⊕Ys; letMhdenote the order ofYh,1hs. LetD=D
Γ, (aij)1i,jθ, (gi)1iθ, (χj)1jθ,(λij)1i<jθ, ij
be a linking datum. We denote byu(D)the algebra presented by generatorsa1, . . . , aθ,y1, . . . , ysand relations
yMhh= 1, ymyh=yhym, for all1m, hs, (5.8)
(1.4), (1.5), (1.6) and
aNαI= 0, for allα∈Φ+I, I∈ X. (5.9)
Remark 5.10. – In the preceding definition, one could consider only linking data withλij= 1 or 0. Indeed, one can replace the generatorai byλ−ij1aiwheneverλij= 0for somejwhich is unique by Lemma 5.6. The other relations do not change since they are homogeneous in theai’s.
However, in the more general case where the relations (5.9) have a non-zero right side, one needs general linking data.
Example 5.11. – Here is a linking datum where all the connected components are points:
• · · · •
• •
• · · · •
Example 5.12. – Let B:= (bij)1i,jR be a finite Cartan matrix,0M R and q∈k a root of unity of orderN; we assume N is odd, and prime to 3if B contains a component of type G2. Let d1, . . . , dR be integers in {1,2,3}such that dibij =djbji. Let θ=R+M, B := (bij)1i,jM andA= (aij)be the Cartan matrix
A=
B 0 0 B
.
LetΓ = (Z/(N))R,g1, . . . , gR the canonical basis ofΓandχ1, . . . , χR be the character given byχi(gj) =qdibij; letgR+j=gj,χR+j=χ−j1,1jM. Note thatjandj+Rare linkable, 1jM. Finally, letλj,j+R= 1if1jMand 0 otherwise; then(λij)1i<jθis a linking datum for Γ, (aij), g1, . . . , gθ andχ1, . . . , χθ. The Hopf algebra u(D) with comultiplication determined by (1.9) is the parabolic part of a Frobenius–Lusztig kernel. Since the numeration of the Dynkin diagram is so far arbitrary, any such parabolic appears in this way.
Example 5.13. – Here are some exotic examples of linking data:
Take 4 copies of A3 and label the vertices such that {1,2,3}, {4,5,6}, {7,8,9} and {10,11,12}are the connected components. Then link3with4,6with7,9with10and12with 1. It is possible to realize this linking overZ/(N)12for any oddN; the corresponding braiding will be symmetric in each component, that is, the corresponding subalgebra is the “Borel part”
of a Frobenius–Lusztig kernel. More examples arise considering more copies of more general components.
See [13] for a combinatorial description of all linkings of Dynkin diagrams.
